We will give generalized definitions called type II n-cocycles and weak quasi-bialgebra and also show properties of type II n-cocycles and some results about weak quasi-bialgebras, for instance, construct a new structure of tensor product algebra over a module algebra on weak quasi-bialgebras.

1. Introduction

We will introduce a new generalized definition called type II n-cocycle; the namely, relax the invertible condition of associator in n-cocycle up to adding an associator satisfies all forms in definition of n-cocycle together with the original associator, and both need not to be invertible for each other; then we give examples to illustrate it clearly. Majid have shown many results about n-cocycle in [1], and we obtain some results including cohomologous concept through this new definition, main properties of type II n-cocycles, and its simple classification.

It is well known that quasi-bialgebras and quasi-Hopf algebras play important roles in quantum group theory, and these concepts were introduced by Drinfel’d in [2] who relaxed the coassociative law of Δ up to conjugation. In this paper, we will show a new definition called weak quasi-bialgebra, a generalization of quasi-bialgebras, and there are simple examples to illustrate. Authors show results for weak quasi-bialgebras in place of quasi-bialgebras (cf. [1, 3]), including that there exists an algebra structure on A⊗H, a generalization of the algebra product in [3], where H is a weak quasi-bialgebra and A is a H-module algebra.

We follow all the notation and conventions in [1], throughout the paper. In the following, k will be a fixed field throughout, and all algebras, coalgebras, vector spaces, and so forth are over k automatically unless specified. We recall definitions as follows.

Definition 1.1.

For any bialgebra, if there is an invertible element θ∈H⊗n such that
(1.1)∂θ=(Δ0θΔ2θ⋯Δsθ)(Δ1θΔ3θ⋯Δtθ)=1,
where integers s and t are max even number and max odd number, respectively, in {0,1,…,n}, we call θ an n-cocycle. If ϵi(θ)=1 (0⩽i⩽n), then the cocycle is counital. We define Δi=id⊗⋯⊗Δ⊗⋯⊗id, ϵi=id⊗⋯⊗ϵ⊗⋯⊗id(1⩽i⩽n),Δ0=1⊗θ, and Δn+1=θ⊗1.

Definition 1.2.

Let H be a k-algebra with unit and homomorphisms Δ:H→H⊗H, ϵ:H→k. If there exists a counital 3-cocycle Φ∈H⊗3 rendering that, for all h∈H,
(1.2)(id⊗Δ)Δh=Φ(Δ⊗id)(Δh)Φ-1,(ϵ⊗id)Δh=1k⊗h,(id⊗ϵ)Δh=h⊗1k.
Then H is called a quasi-bialgebra together with coproduct Δ and counit ϵ, and call Φ an associator.

We will denote the tensor components of Φ,ϕ with big and small letters, respectively, for instance,
(1.3)Φ=∑X1⊗X2⊗X3=∑Y1⊗Y2⊗Y3=∑Z1⊗Z2⊗Z3etc.,ϕ=∑x1⊗x2⊗x3=∑y1⊗y2⊗y3=∑z1⊗z2⊗z3etc.,
where xi is the ith factor.

Definition 1.3.

Let H be a quasi-bialgebra and A a vector space. If A has a multiplication and the unit 1A obeying that
(1.4)a(bc)=∑(X1·a)[(X2·b)(X3·c)],h·(ab)=∑(h1·a)(h2·b),h·1A=ϵ(h)1A,
for any a,b,c∈A and h∈H, where ·:<H⊗A→A is the H-module structure map of A, then say A is a left H-module algebra.

In bialgebras, the composition of any two algebra homomorphisms Δi,Δj satisfies the equality ΔiΔj=Δj+1Δi for all i≤j. We will use this equation frequently.

2. Type II Cocycles and Weak Quasi-BialgebrasDefinition 2.1.

An associative algebra H with unit over a commutative ring R called a fake bialgebra, if there are two algebra homomorphisms Δ:H→H⊗H and ϵ:H→R.

Definition 2.2.

Let H be a fake bialgebra, and elements Φ,ϕ∈H⊗n. Denote
(2.1)∂(Φ,ϕ)=(Δ0ΦΔ2Φ⋯ΔsΦ)(Δ1ϕΔ3ϕ⋯Δtϕ),χ(Φ,ϕ)=(ΔtΦΔt-2Φ⋯Δ1Φ)(ΔsϕΔs-2ϕ⋯Δ0ϕ),
where integers s and t in {0,1,…,n} are max even number and max odd number, respectively. If there exists an element ψ(Φ,ϕ)∈H⊗n+1 with ψ(Φ,ϕ)∂(Φ,ϕ)=χ(Φ,ϕ), which makes ∂(Φ,ϕ) and Φ, ϕ satisfy all equalities that one side of equalities is not a single item at least, similar to (1.1). Then call the pair (Φ,ϕ) a type II n-cocycle for H and denote it by (Φ,ϕ)nII. The cocycle (Φ,ϕ)nII is called counital, if both Φ and ϕ are counital.

There is a nature way to define type I n-cocycle (Φ,ϕ)nI similarly. If we require the type II n-cocycle (Φ,ϕ)nII for H to satisfy all transformations of (1.1), but each side of formulas must have one item at least, then the type II n-cocycle (Φ,ϕ)nII is called type I n-cocycle and denoted by (Φ,ϕ)nI.

We write ∂, ψ, and χ briefly for ∂(Φ,ϕ), ψ(Φ,ϕ), and χ(Φ,ϕ) without confusions, respectively. To clarify a new definition above, we give simple examples on a fake bialgebra H. In the following, we discuss type II n-cocycles (Φ,ϕ)nII only.

Example 2.3.

Cocycle (Φ,ϕ)1II means both Φ and ϕ are in H, obeying that
(2.2)∂Δ1Φ=Δ0ΦΔ2Φ,Δ0ϕ∂=Δ2ΦΔ1ϕ.
And there is ψ∈H⊗2 such that
(2.3)Δ2ϕΔ0ϕ=Δ1ϕψ,Δ1ΦΔ2ϕ=ψΔ0Φ,
where ∂=Δ0ΦΔ2ΦΔ1ϕ.

Example 2.4.

Cocycle (Φ,ϕ)3II and ∂=Δ0ΦΔ2ΦΔ4ΦΔ1ϕΔ3ϕ, ψ∈H⊗4, where Φ,ϕ∈H⊗3, satisfy that
(2.4)∂Δ3ΦΔ1Φ=Δ0ΦΔ2ΦΔ4Φ,∂Δ3ΦΔ1ΦΔ4ϕ=Δ0ΦΔ2Φ,∂Δ3Φ=Δ0ΦΔ2ΦΔ4ΦΔ1ϕ,(Δ0ϕ)∂Δ3ΦΔ1Φ=Δ2ΦΔ4Φ,etc.,Δ3ΦΔ1Φ=ψΔ0ΦΔ2ΦΔ4Φ,Δ3ΦΔ1ΦΔ4ϕ=ψΔ0ΦΔ2Φ,Δ3ΦΔ1ΦΔ4ϕΔ2ϕ=ψΔ0Φ,Δ1ΦΔ4ϕΔ2ϕ=(Δ3ϕ)ψΔ0Φ,etc.

Observing examples, we can see that ∂ and Φ are replaced by ψ and ϕ, respectively, after moving to a corresponding place in the other side of equations, and vice versa. Obviously, n-cocycle θ must be a type II 2-cocycle (θ,θ-1)nII and ∂(θ,θ-1)=ψ(θ,θ-1)=1.

Example 2.5.

Let A be an associative algebra with an idempotent q∈A over a field k. Define Δ:A→A⊗A by Δ(a)=a⊗a and ϵ:A→k by ϵ(a)=0k, for all a∈A. It is clear that A is a fake bialgebra. We set Φ=ϕ=q and ψ=q⊗q, then ∂=χ=q⊗q. It is easy to check that (Φ,ϕ) is a type II 1-cocycle.

Proposition 2.6.

Let (Φ,ϕ)nII be a cocycle for a fake bialgebra H, and denote
(2.5)∂=(Δ0ΦΔ2Φ⋯ΔsΦ)(Δ1ϕΔ3ϕ⋯Δtϕ),χ=(ΔtΦΔt-2Φ⋯Δ1Φ)(ΔsϕΔs-2ϕ⋯Δ0ϕ),
where integers s and t in {0,1,…,n} are max even number and max odd number, respectively. Then one has the following.

χ=ψΔ0(Φϕ)=Δt(Φϕ)ψ=(ΔtΦΔt-2Φ⋯Δ1Φ)(Δ1ϕΔ3ϕ⋯Δtϕ)=χ∂=χ2, and ∂=Δ0(Φϕ)∂=∂Δt(Φϕ)=(Δ0ΦΔ2Φ⋯ΔsΦ)(ΔsϕΔs-2ϕ⋯Δ0ϕ)=∂χ=∂2.

If ∂ is commutative with χ, then χ=∂. Especially, if either ∂ or χ is zero, then the other one is zero too. On the other hand, if either of elements ∂ and χ is not zero, then the rest elements in set {∂,χ,ψ} are not zero.

If Δ0(Φϕ)-1 is a left unit and Φ is not a right zero divisor, then ∂=χ=ψ=0.

If Φ has a right inverse ΦR-1, so do ∂, ψ, and χ. Similarly, if ϕ has a left inverse ϕL-1, so do ∂, ψ, and χ.

Proof.

(1) We obtain that χ=Δt(Φϕ)ψ by
(2.6)(Δtϕ)ψ=Δt-2Φ⋯Δ3ΦΔ1ΦΔsϕ⋯Δ2ϕΔ0ϕ
and χ=ψΔ0(Φϕ) by
(2.7)ψΔ0Φ=ΔtΦ⋯Δ3ΦΔ1ΦΔsϕ⋯Δ4ϕΔ2ϕ,
since Δi is a homomorphism. And
(2.8)χ=ψ∂=(ψΔ0Φ⋯ΔsΦ)Δ1ϕ⋯Δtϕ=(ΔtΦ⋯Δ1Φ)(Δ1ϕ⋯Δtϕ).
Analogously, we have that
(2.9)Δ0(Φϕ)∂=Δ0Φ((Δ0ϕ)∂)=Δ0Φ(Δ2Φ⋯ΔsΦΔ1ϕ⋯Δtϕ)=∂,∂Δt(Φϕ)=(∂ΔtΦ)Δtϕ=(Δ0Φ⋯ΔsΦΔ1ϕ⋯Δt-2ϕ)Δtϕ=∂.
Then, we get easily that χ=ψ∂=ψΔ0(Φϕ)∂=χ∂ and
(2.10)∂χ=(∂ΔtΦΔt-2Φ⋯Δ1Φ)Δ1ϕ⋯Δt-2ϕΔtϕ=Δ0ΦΔ2Φ⋯ΔsΦΔ1ϕ⋯Δt-2ϕΔtϕ=∂.
Finally, there is the equality that
(2.11)∂=∂χ=(∂ΔtΦΔt-2Φ⋯Δ1Φ)(ΔsϕΔs-2ϕ⋯Δ0ϕ)=(Δ0ΦΔ2Φ⋯ΔsΦ)(ΔsϕΔs-2ϕ⋯Δ0ϕ).
We, last, compute that
(2.12)∂2=(∂χ)∂=∂(χ∂)=∂χ=∂,χ2=(χ∂)χ=χ(∂χ)=χ∂=χ.
Therefore ∂ and χ are idempotent.

(2) Obviously, we get this by statement (1). Let ∂≠0 and assume χ=0, from the previous part, that yields to ∂=0 contradicting ∂≠0. Therefore χ must be zero, and ψ=0 for the same reason.

(3) The equality (Δ0(Φϕ)-1)∂=0 suggests that ∂=0, and then χ=0. It is clear that Δ0Φ⋯ΔsΦ=0 because ∂ΔtΦ⋯Δ1Φ=Δ0Φ⋯ΔsΦ. In addition, ΔtΦ⋯Δ3ΦΔ1Φ=ψΔ0ΦΔ2Φ⋯ΔsΦ=0 implies that
(2.13)ψΔ0Φ=ΔtΦ⋯Δ3ΦΔ1ΦΔsϕ⋯Δ4ϕΔ2ϕ=0.
As a result, we have ψ=0 if Φ is not a right zero divisor.

(4) It is easy to obtain that ∂ΔtΦ⋯Δ3ΦΔ1ΦΔsΦR-1⋯Δ2ΦR-1Δ0ΦR-1=1 and ψΔ0Φ⋯ΔsΦΔ1ΦR-1⋯ΔtΦR-1=1 as ∂ΔtΦ⋯Δ1Φ=Δ0Φ⋯ΔsΦ and ψΔ0Φ⋯ΔsΦ=ΔtΦ⋯Δ1Φ, respectively. But then χ=ψ∂ and χ has a right inverse. Likewise, we can prove the rest part.

Furthermore, if ∂ or χ has a one-side inverse, it makes sense that ∂=χ=ψ=1 since both ∂ and χ are idempotent. We also have that Φϕ=1 which indicates Φ is a left unit and ϕ a right unit, by Δ0(Φϕ)∂=∂ if ∂=1. Hence ∂ and χ cannot be anything but the identity element if one of them is a one-side unit.

Corollary 2.7.

The following statements are equivalent.

Φ has a right inverse.

∂=χ=ψ=1.

ϕ has a left inverse.

∂ is a one-side unit.

χ is a one-side unit.

ψ has a left inverse.

Proof.

(Sketch of Proof).

Check by (1)⇒(2)⇒(3)⇒(4)⇒(5)⇒(6)⇒(1).

Equality (Δ0(Φϕ)-1)∂=0 suggests that classification of ∂ is divided into three types. The first type ∂=0, and the second ∂=1 if Δ0(Φϕ)=1, that is, Φϕ=1. The last one is that ∂ is a right zero divisor.

Example 2.8.

In algebra ℤ6 over the integer ring ℤ, we define Δ:ℤ6→ℤ6⊗ℤ6 given by Δ3-=3-⊗3- and Δx-=x-⊗3- for any x-∈ℤ6-{3-}, and ϵ:ℤ6→ℤ by ϵ(y-)=y for all y-∈ℤ6 such that (ℤ6,Δ,ϵ) is a fake bialgebra. Set Φ=ϕ=3-∈ℤ6 such that ∂=3-⊗3-. The product of any two elements in the set {ΔiΦ,∂}(i=0,1,2) equals 3-⊗3-, obviously. We also set ψ=∂; then, it is easy to prove that (Φ,ϕ)1II is a cocycle and a right zero-divisor ∂.

Proposition 2.9.

Let H be a fake bialgebra with counital law of ϵ. If Φ,ϕ∈H and ϵ(Φ)=1 (ϵ(ϕ)=1, resp.), then ∂(Φ,ϕ) (χ(Φ,ϕ), resp.) is counital if and only if Φϕ=1 (ϕΦ=1, resp.).

Proof..

Since that ∂(Φ,ϕ)=Δ0ΦΔ2ΦΔ1ϕ=(Φ⊗Φ)Δϕ, we have ϵi∂(Φ,ϕ)=ϵi(Φ⊗Φ)ϵiΔϕ=ϵ(Φ)Φϕ=Φϕ rendering that ϵi∂(Φ,ϕ)=1 if and only if Φϕ=1, where i=1,2.

Proposition 2.10.

Let H be a fake bialgebra with coassociative law of Δ and there are elements Φ,ϕ in H. If ∂(Φ,ϕ) obeys ∂Δ1Φ=Δ0ΦΔ2Φ, then Δ3∂Δ1∂=Δ0∂Δ2∂. Especially, ∂ is a 2-cocycle if ∂ is invertible if H is a bialgebra.

Proof..

To obtain the result, we observe that
(2.14)Δ3∂Δ1∂=(∂⊗1)(Δ⊗id)∂=(∂⊗1)(Δ⊗id)((Φ⊗Φ)Δ1ϕ)=(∂⊗1)(ΔΦ⊗Φ)(Δ1Δ1ϕ)=(∂⊗1)(ΔΦ⊗Φ)(Δ2Δ1ϕ)=(∂ΔΦ⊗Φ)(Δ2Δ1ϕ)=(Φ⊗Φ⊗Φ)Δ2Δ1ϕ=(1⊗∂)(Φ⊗ΔΦ)Δ2Δ1ϕ=(1⊗∂)Δ2(Φ⊗Φ)Δ2Δ1ϕ=Δ0∂Δ2∂.

We have known that θγ=(γ⊗γ)θΔγ-1 is cohomologous to θ for a bialgebra H if θ is a counital 2-cocycle, which was mentioned by Majid in [1]. Let H be a bialgebra, Φ,ϕ∈H and cocycle (σ,δ)2II for H. Denote that σ(Φ,ϕ)=Δ0ΦΔ2ΦσδσΔ1ϕ and δ(Φ,ϕ)=Δ1ΦδσδΔ2ϕΔ0ϕ. Then we have the following.

Proposition 2.11.

If equality ∂(σ,δ)Δ2δΔ0δΔ3σ=Δ1δ holds and ∂(σ,δ) is a commutative element in set {Δ1σ,Δ3δ,Δ3σ}, and 1⊗Δ(ϕΦ)(=Δ(ϕΦ)⊗1) commutes with any element in set {Δ1σ,Δ1δ,Δ2σ,Δ2δ} as well, then
(2.15)Δ0σ(Φ,ϕ)Δ2σ(Φ,ϕ)=Δ3σ(Φ,ϕ)Δ1σ(Φ,ϕ).

Proof.

A long equality showed that
(2.16)Δ0σ(Φ,ϕ)Δ2σ(Φ,ϕ)=(1⊗(Φ⊗Φ)σδσΔϕ)(id⊗Δ)(σδσΔϕ)=(Φ⊗Φ⊗Φ)Δ0(σδσ)Δ0Δ1ϕΔ2Δ0ΦΔ2(σδσ)Δ2Δ1ϕ=(Φ⊗Φ⊗Φ)Δ0(σδσ)Δ2Δ0(ϕΦ)Δ2(σδσ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ0(σδσ)Δ2(σδσ)Δ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ0(σδ)Δ0σΔ2σΔ2δΔ2σΔ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ0(σδ)∂(σ,δ)Δ3σΔ1σΔ2δΔ2σΔ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)∂(σ,δ)Δ3σΔ3δψ(σ,δ)Δ0σΔ2σΔ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ3(σδ)∂(σ,δ)ψ(σ,δ)∂(σ,δ)Δ3σΔ1σΔ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ3(σδ)∂(σ,δ)Δ3σΔ1σΔ2δΔ0δΔ3σΔ1σΔ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ3(σδσ)Δ1σ∂(σ,δ)Δ2δΔ0δΔ3σΔ1σΔ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)Δ3(σδσ)Δ1(σδσ)Δ1Δ2(ϕΦ)Δ1Δ1ϕ=(Φ⊗Φ⊗Φ)(σδσ⊗1)(Δ(ϕΦ)⊗1)(Δ⊗id)(σδσΔϕ)=((Φ⊗Φ)σδσΔϕ⊗1)(Δ⊗id)((Φ⊗Φ)σδσΔϕ)=Δ3σ(Φ,ϕ)Δ1σ(Φ,ϕ).

There exists a similar version for δ(Φ,ϕ), namely, the following preposition.

Proposition 2.12.

If there is the equation Δ1δΔ3δψ(σ,δ)Δ0σ=Δ2δ and ∂(σ,δ) commutes with any element in set {Δ0σ,Δ0δ}, and 1⊗Δ(ϕΦ)(=Δ(ϕΦ)⊗1) is a commutative element in set {Δ0σ,Δ0δ,Δ3σ,Δ3δ} as well, then
(2.17)Δ1δ(Φ,ϕ)Δ3δ(Φ,ϕ)=Δ0δ(Φ,ϕ)Δ2δ(Φ,ϕ).

Proposition 2.13.

Let H be a bialgebra and (Φ,ϕ)2II a counital cocycle for H, and define Δ(Φ,ϕ)(h)=ΦϕΦΔ(h)ϕΦϕ for all h∈H, then the algebra H with original ϵ and Δ(Φ,ϕ) consists a new coalgebra if χΔ3ΦΔ1Φ=Δ0ΦΔ2Φ. Moreover, Δ(Φ,ϕ) is an algebra map if (ϕΦ)3=1, then algebra H is a bialgebra with comultiplication Δ(Φ,ϕ).

Proof.

It is clear that (ϵ⊗id)Δ(Φ,ϕ)(h)=(id⊗ϵ)Δ(Φ,ϕ)(h)=h. So we only need to show the coassociative law of Δ(Φ,ϕ). For all h∈H, we obtain
(2.18)(Δ(Φ,ϕ)⊗id)Δ(Φ,ϕ)(h)=(Δ(Φ,ϕ)⊗id)(ΦϕΦΔ(h)ϕΦϕ)=Δ3(ΦϕΦ)Δ1(ΦϕΦ)Δ1Δ1(h)Δ1(ϕΦϕ)Δ3(ϕΦϕ)=Δ3(Φϕ)ψΔ0ΦΔ2ΦΔ1ϕΔ1ΦΔ2Δ1(h)Δ1ϕΔ1ΦΔ2ϕΔ0ϕ∂Δ3(Φϕ)=χ∂Δ3ΦΔ1ΦΔ2Δ1(h)Δ1ϕΔ3ϕψ∂=χΔ3ΦΔ1ΦΔ2Δ1(h)Δ1ϕΔ3ϕχ=∂Δ3ΦΔ1ΦΔ2Δ1(h)Δ2ϕΔ0ϕ∂χ=∂ψΔ0ΦΔ2ΦΔ2Δ1(h)Δ2ϕΔ2ΦΔ1ϕΔ3ϕχ=Δ0(Φϕ)∂Δ3ΦΔ1ΦΔ2ΦΔ2Δ1(h)Δ2ϕΔ2ΦΔ1ϕΔ3ϕψΔ0(Φϕ)=Δ0(Φϕ)Δ0ΦΔ2ΦΔ2(ϕΦ)Δ2Δ1(h)Δ2(ϕΦ)Δ2ϕΔ0ϕΔ0(Φϕ)=Δ0(ΦϕΦ)Δ2(ΦϕΦ)Δ2Δ1(h)Δ2(ϕΦϕ)Δ0(ϕΦϕ)=Δ0(ΦϕΦ)(id⊗Δ)(ΦϕΦΔ(h)ϕΦϕ)Δ0(ϕΦϕ)=(id⊗Δ(Φ,ϕ))Δ(Φ,ϕ)(h).
Finally, for any g∈H,
(2.19)Δ(Φ,ϕ)(hg)=ΦϕΦΔ(hg)ϕΦϕ=ΦϕΦΔ(h)ϕΦϕΦϕΦΔ(g)ϕΦϕ=Δ(Φ,ϕ)(h)Δ(Φ,ϕ)(g).

Definition 2.14.

Let (H,Δ,ϵ) be a fake bialgebra. If there exists a cocycle (Φ,ϕ)3II for H obeying that
(2.20)(id⊗Δ)Δ(h)Φ=Φ(Δ⊗id)Δ(h),ϕ(id⊗Δ)Δ(h)=(Δ⊗id)Δ(h)ϕ,(id⊗ϵ)Δ(h)=(ϵ⊗id)Δ(h)=h,
for all h∈H, then H is called a weak quasi-bialgebra.

Example 2.15.

Let H be an associate algebra over field k, where the characteristic of k is not 2. And H is a 4-dimensional vector space with basis {1,i,j,ij} obeying that i2=i,j2=j, and ij=ji. We define homomorphisms Δ:H→H⊗H given by Δ(i)=i⊗j,Δ(j)=j⊗1 and ϵ:H→k given by ϵ(i)=ϵ(j)=0. Obviously, H is a fake bialgebra. Set Φ=j⊗j⊗j and ψ=j⊗1⊗1⊗j, then (Φ,Φ)3II is a cocycle with holding ∂=χ=j⊗j⊗j⊗j. It is routine to check (H,Δ,ϵ,(Φ,Φ)3II) is a weak quasi-bialgebra.

We relax Definition 1.3 by setting that H is a weak quasi-bialgebra so that we can define an algebra structure on A⊗H, if A is a left H-module algebra and H a weak quasi-bialgebra, given by
(2.21)(a#h)(b#g)=∑(y1X1x1·a)(y2X2x2h1·b)#y3X3x3h2g
for all a,b∈A,h,g∈H, while a#h is equal to a⊗h here.

Theorem 2.16.

Let H be a weak quasi-bialgebra and A a left H-module algebra. Then A#H is an associative algebra under the multiplication mentioned above and 1A#1H is the unit.

Proof..

For all a,b,andc∈A and h,g,andl∈H, we easily get that, by properties of ϵ,
(2.22)(1#1)(a#h)=(y1X1x1·1)(y2X2x2·a)#y3X3x3h=ϵ(y1X1x1)(y2X2x2·a)#y3X3x3h=a#h,(a#h)(1#1)=(y1X1x1·a)(y2X2x2h1·1)#y3X3x3h2=(y1X1x1·a)ϵ(y2X2x2h1)#y3X3x3h2=a#h.
Now we show the associative law:
(2.23)[(a#h)(b#g)](c#l)=[(y1X1x1·a)(y2X2x2h1·b)#y3X3x3h2g](c#l)=(w1Y1z1·(y1X1x1·a)(y2X2x2h1·b))×(w2Y2z2(y3X3x3h2g)1·c)#w3Y3z3(y3X3x3h2g)2l(2.24)=((w11Y11z11y1X1x1·a)(w12Y12z12y2X2x2h1·b))×(w2Y2z2y31X31x31h21g1·c)#w3Y3z3y32X32x32h22g2l.
But
(2.25)Δ1ϕΔ1ΦΔ1ϕΔ3ϕΔ3ΦΔ3ϕ=Δ1ϕΔ1ΦΔ4ϕΔ2ϕ(Δ0ϕ)∂Δ3ΦΔ3ϕ=Δ1ϕΔ1ΦΔ4ϕΔ2ϕ(Δ0ϕ)∂=Δ1ϕ(Δ3ϕ)ψ∂,
and then we obtain that (2.23) is
(2.26)((z11x1ψ1∂1·a)(z12x2ψ2∂2h1·b))(z2x31ψ3∂3h21g1·c)#z3x32ψ4∂4h22g2l=(Y1z11x1ψ1∂1·a)((Y2z12x2ψ2∂2h1·b)(Y3z2x31ψ3∂3h21g1·c))#z3x32ψ4∂4h22g2l.
On the other hand, the equation
(2.27)Δ4ΦΔ1ϕ(Δ3ϕ)ψ∂=Δ2ϕ(Δ0ϕ)∂ψ∂=Δ2ϕ(Δ0ϕ)∂χ=Δ2ϕ(Δ0ϕ)∂ψΔ0ΦΔ0ϕ=Δ2ϕΔ2ΦΔ4ΦΔ1ϕ(Δ3ϕ)ψΔ0ΦΔ0ϕ=Δ2ϕΔ2ΦΔ2ϕΔ0ϕΔ0ΦΔ0ϕ
makes (2.26) equal
(2.28)(w1Y1z1·a)((w21Y21z21y1X1x1h1·b)(w22Y22z22y2X2x2h21g1·c))#w3Y3z3y3X3x3h22g2l=(w1Y1z1·a)((w21Y21z21h11y1X1x1·b)(w22Y22z22h12y2X2x2g1·c))#w3Y3z3h2y3X3x3g2l=(w1Y1z1·a)(w2Y2z2h1·((y1X1x1·b)(y2X2x2g1·c)))#w3Y3z3h2y3X3x3g2l=(a#h)[(b#g)(c#l)].
Hence, [(a#h)(b#g)](c#l)=(a#h)[(b#g)(c#l)].

If ϕ is an inverse of Φ, then the multiplication becomes that
(2.29)(a#h)(b#g)=∑(x1·a)(x2h1·b)#x3h2g,
which is as exact as the one in [3].

Acknowledgments

The authors were supported by Guangxi Science Foundation (2011GXNSFA01844), the Scientific Research Foundation of Guangxi Educational Committee (200911 MS145), and 2012 Guangxi New Century Higher Education Teaching Reform Project (2012JGA162). The authors would like to thank the referee for precious suggestions.

MajidS.Drinfel'dV. G.Quasi-Hopf algebrasBulacuD.PanaiteF.Van OystaeyenF.Quasi-Hopf algebra actions and smash products